**APPENDIX**

**PHYSICAL BACKGROUND**

This short appendix contains the physical toolbox used throughout the book.

**A.1 Springs**

The toolbox for our thought experiments includes two types of springs: linear springs and constant tension springs.

**Zero-length springs.** A zero-length spring is the one whose tension is directly proportional to its length: stretching such a string to a length *x* requires the force *kx*. Here *k* is a constant (called the Hooke’s constant) that characterizes a particular spring. A small *k* means a lax spring, while a large *k* means a stiff spring. Note that the unstretched length of such a spring is zero.

**POTENTIAL ENERGY OF A ZERO-LENGTH SPRING.** By the definition, the potential energy of a spring is the work required to stretch the spring ...